Download MSE-630 Week 3

MSE-630 Week 3
Conductivity, Energy Bands and
Charge Carriers in
Semiconductors
Objectives:
• To understand conduction, valence energy
bands and how bandgaps are formed
• To understand the effects of doping in
semiconductors
• To use Fermi-Dirac statistics to calculate
conductivity and carrier concentrations
• To understand carrier mobility and how it is
influenced by scattering
• To introduce the idea of “effective mass”
• To see how we can use Hall effect to determine
carrier concentration and mobility
Conductivity
Charge carriers follow a
random path unless an
external field is applied.
Then, they acquire a drift
velocity that is dependent
upon their mobility, mn and the
strength of the field, x
Vd = -mn x
The average drift velocity, vav is dependent
Upon the mean time between collisions, 2t
Charge Flow and Current Density
Current density, J, is the rate at
which charges, cross any plane
perpendicular to the flow direction.
J = -nqvd = nqmnx = sx
n is the number of charges, and
-19
q is the charge (1.6 x 10
C)
The total current density depends upon the total charge
carriers, which can be ions, electrons, or holes
J = q(nmn + pmp) x
OHM’s Law:
V = IR
Resistance, R(W) is an extrinsic quantity. Resistivity, r(Wm), is the
corresponding intrinsic property.
r = R*A/l
Conductivity, s, is the reciprocal of resistivity: s(Wm)-1 = 1/r
When we add carriers by doping, the number of additional carrers, Nd,
far exceeds those in an intrinsic semiconductor, and we can treat
conductivity as
s = qNdmd
In general, np=ni2, where ni is the intrinsic concentration of carriers at a
given temperature (ni = 1.5 x 1010 cm-3 in Si) In a doped semiconductor,
charges balance, thus Na+n = Nd+p. For an n-type semiconductor, n ~ Nd,
and p ~ ni2/Nd
As the distance between
atoms decreases, the
energy of each orbital
must split, since
according to Quantum
Mechanics we cannot
have two orbitals with
the same energy.
The splitting results in “bands” of
electrons. The energy difference
between the conduction and valence
bands is the “gap energy” We must
supply this much energy to elevate an
electron from the valence band to the
conduction band. If Eg is < 2eV, the
material is a semiconductor.
In the ground state, at 0K,
all the electrons have
energy less than Ef, the
Fermi energy. If we add
energy we can boost an
electron into the conduction
band. If we add dopants,
we can enhance the
number of positive or
negative carriers, vastly
increasing conductivity
Adding a column 5 atom to Si
(e.g., As) adds a 5th, loosely
bound electron. The ionization
energy is very low, and it can
be boosted to the conduction
band easily. This is an
electron donor
If we add a material that is short
one electron (e.g., B, valence
3) we will create an electron
“sink” in the material. This
results in “holes” that migrate
in the valence band. Holes
carry positive charge.
In an intrinsic semiconductor,
elevating an electron to the
conduction band leaves behind
a hole in the valence band, or
creates an “electron-hole pair”,
or EHP
Fermi-Dirac statistics
•
Conductivity in semiconductors is
described by Fermi-Dirac
statistics. In an intrinsic
semiconductor, the conductivity is:
For an extrinsic semiconductor,
corresponding probability of an
electron is
s = s oe

E
kT
Where E = Ec-Ed for an n-type,
and E = Ea-Ev for a p-type
s = s oe
 Eg
2 kT
At low temperatures, the
conduction is provided by the
dopants, and the slope is
defined by E/k. When the
available electrons/holes is
depleted in the “exhaustion”
range, the slope is constant.
When the temperature
becomes high enough,
energies exceed Eg, and the
semiconductor moves into
the intrinsic conduction
range, with a slope of Eg/2k.
Effective Mass
In general, the curve of Energy vs. k is nonlinear, with E increasing as k increases.
E = ½ mv2 = ½ p2/m = h2/4pm k2
We can see that energy varies inversely with
mass. Differentiating E wrt k twice, and
solving for mass gives:
2
h
m =
2
d E
2p
2
dk
*
Effective mass is significant because it
affects charge carrier mobility, and
must be considered when calculating
carrier concentrations or momentum
Effective mass and other semiconductor properties may be found in
Appendix III
In reality, band structures are highly
dependent upon crystal orientation. This
image shows us that the lowest band gap
in Si occurs along the [100] directions, whil
for GaAs, it occurs in the [111]. This is why
crystals are grown with specific
orientations.
The diagram showing the
constant energy surface
(3.10 (b)), shows us that
the effective mass varies
with direction. We can
calculate average effective
mass from:
1 1 1
2 
=   
*
mn 3  ml mt 
Hall Effect
When a magnet field is appplied
perpendicular to the direction in
which a charged particle
(electron or hole) is moving, the
particle will be deflected as
shown
The force on the particle will be
F = q(x+vXB)
In the x-direction, the force will
be Fy = q(xy+vxXBz)
To counter the flow of particles in
the x-direction, we apply a field
xy=vxBz so that the net force is
zero. The applied field is called
the Hall effect, and the resulting
voltage, VH=xyd
Drift velocity for an electron in the x-direction is:
<vy>=-Jy/qn
where J is the current density, n is the number of carriers
and q is the charge
Defining the Hall coefficient, RH=1/qn, then
 xy = vxBz = - Jx/qn Bz = RHJxBz
 and
 I x 
J x Bz  wt  z
Ixz
1
n=
=
=
=
qRH
qx y
qVAB / w qtVAB
Measuring the resistance gives the resistivity:
r (Wm)= Rwt/L = (Vcd/Ix)/(L/wt)
Since conductivity, s = 1/r = qmnn, the mobility is:
mn = s/qn = (1/r)/q(1/qRH) = RH/r